Carlson-Simpson's lemma and applications in reverse mathematics

TL;DR

This paper explores the reverse mathematical strength of Carlson-Simpson's infinitary theorems, demonstrating their applications in Ramsey theory and establishing the Open Dual Ramsey theorem within a specific logical framework.

Contribution

It provides a reverse mathematics analysis of Carlson-Simpson's theorems and proves the Open Dual Ramsey theorem in .

Findings

Open Dual Ramsey theorem holds in 

Finite big Ramsey numbers for triangle-free graphs established

Connections between Carlson-Simpson's theorems and reverse mathematics clarified

Abstract

We study the reverse mathematics of infinitary extensions of the Hales-Jewett theorem, due to Carlson and Simpson. These theorems have multiple applications in Ramsey's theory, such as the existence of finite big Ramsey numbers for the triangle-free graph, or the Dual Ramsey theorem. We show in particular that the Open Dual Ramsey theorem holds in $\mathsf{ACA}^{+}_0$.